The Nambu–Goto action is the simplest invariant
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
in
bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum.
In the 1980s, supersymmetry was discovered in the co ...
, and is also used in other theories that investigate string-like objects (for example,
cosmic string
Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a symmetry-breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking was not simpl ...
s). It is the starting point of the analysis of zero-thickness (infinitely thin) string behavior, using the principles of
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...
. Just as the action for a free point particle is proportional to its
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
— ''i.e.'', the "length" of its world-line — a relativistic string's action is proportional to the area of the sheet which the string traces as it travels through spacetime.
It is named after Japanese physicists
Yoichiro Nambu
was a Japanese-American physicist and professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded half of the Nobel Prize in Physics in 2008 for the discovery in 1960 of the mechanism ...
and
Tetsuo Goto.
Background
Relativistic Lagrangian mechanics
The basic principle of Lagrangian mechanics, the
principle of stationary action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
, is that an object subjected to outside influences will "choose" a path which makes a certain quantity, the ''action'', an extremum. The action is a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
, a mathematical relationship which takes an entire path and produces a single number. The ''physical path'', that which the object actually follows, is the path for which the action is "stationary" (or extremal): any small variation of the path from the physical one does not significantly change the action. (Often, this is equivalent to saying the physical path is the one for which the action is a minimum.) Actions are typically written using Lagrangians, formulas which depend upon the object's state at a particular point in space and/or time. In non-relativistic mechanics, for example, a point particle's Lagrangian is the difference between kinetic and potential energy:
. The action, often written
, is then the integral of this quantity from a starting time to an ending time:
:
(Typically, when using Lagrangians, we assume we know the particle's starting and ending positions, and we concern ourselves with the ''path'' which the particle travels between those positions.)
This approach to mechanics has the advantage that it is easily extended and generalized. For example, we can write a Lagrangian for a
relativistic particle, which will be valid even if the particle is traveling close to the speed of light. To preserve
Lorentz invariance
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation
In physics, the Lorentz transformations are a six-parameter famil ...
, the action should only depend upon quantities that are the same for all (Lorentz) observers, i.e. the action should be a
Lorentz scalar
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation
In physics, the Lorentz transformations are a six-parameter famil ...
. The simplest such quantity is the ''proper time'', the time measured by a clock carried by the particle. According to special relativity, all Lorentz observers watching a particle move will compute the same value for the quantity
:
and
is then an infinitesimal proper time. For a point particle not subject to external forces (''i.e.'', one undergoing inertial motion), the
relativistic action is
:
World-sheets
Just as a zero-dimensional point traces out a world-line on a spacetime diagram, a one-dimensional string is represented by a ''world-sheet''. All world-sheets are two-dimensional surfaces, hence we need two parameters to specify a point on a world-sheet. String theorists use the symbols
and
for these parameters. As it turns out, string theories involve higher-dimensional spaces than the 3D world with which we are familiar; bosonic string theory requires 25 spatial dimensions and one time axis. If
is the number of spatial dimensions, we can represent a point by the vector
:
We describe a string using functions which map a position in the
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for ...
(
,
) to a point in spacetime. For each value of
and
, these functions specify a unique spacetime vector:
:
The functions
determine the shape which the world-sheet takes. Different Lorentz observers will disagree on the coordinates they assign to particular points on the world-sheet, but they must all agree on the total ''proper area'' which the world-sheet has. The Nambu–Goto action is chosen to be proportional to this total proper area.
Let
be the metric on the
-dimensional spacetime. Then,
:
is the
induced metric on the world-sheet, where
and
.
For the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
of the world-sheet the following holds:
:
where
and
Using the notation that:
:
and
:
one can rewrite the
metric :
:
:
the Nambu–Goto action is defined as
:
where
.
The factors before the integral give the action the correct units, energy multiplied by time.
is the tension in the string, and
is the speed of light. Typically, string theorists work in "natural units" where
is set to 1 (along with Planck's constant
and Newton's constant
). Also, partly for historical reasons, they use the "slope parameter"
instead of
. With these changes, the Nambu–Goto action becomes
:
These two forms are, of course, entirely equivalent: choosing one over the other is a matter of convention and convenience.
Two further equivalent forms are
:
and
:
Typically, the Nambu–Goto action does not yet have the form appropriate for studying the quantum physics of strings. For this it must be modified
in a similar way as the action of a point particle. That is classically equal to minus mass times the invariant length in spacetime,
but must be replaced by a quadratic expression with the same classical value.
[See Chapter 19 of
Kleinert's standard textbook on ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 5th edition]
World Scientific (Singapore, 2009)
(also availabl
online
For strings the analog correction is provided by the
Polyakov action, which is classically equivalent to the Nambu–Goto action, but gives the 'correct'
quantum theory. It is, however, possible to develop a quantum theory from the Nambu–Goto action in the
light cone gauge
In theoretical physics, light cone gauge is an approach to remove the ambiguities arising from a gauge symmetry. While the term refers to several situations, a null component of a field ''A'' is set to zero (or a simple function of other variables ...
.
See also
*
Dirac membrane
References
Further reading
* Ortin, Thomas, ''Gravity and Strings'', Cambridge Monographs, Cambridge University Press (2004). .
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String theory